MAXWELL - DIRAC EQUATIONS 27
Proof. Let g i— Vg be the unitary representation of P
0
in Ep defined in Theorem 2.2. Let
I H ^ , be the corresponding Lie algebra representation of p in the space of differential
vectors of V, which according to Theorem 2.2 is EQQ. We write
fcr« = ( # ( / , / ) , # * ) , u = (f,f,a)eE00. (2.18)
Since, by Theorem 2.2, the Hilbert space of
C1
-vectors of V is Ei, we have
(ll«lll + £ H^tilli) * C||«||Bi, u £00, (2.19)
for some constant C.
We shall prove the lemma by giving an upper bound for the Laplacian A# of the
representation V:
±EU = Y, &u = (A
M
(/, / ) , A
D
a) , (2.20a)
xen
where
A M = £ ( # )
2
, A
D
= £ ( | £ )
2
.
(2.20b)
xen xen
According to Theorem 2.2, the "Dirac part" of
T1
is identical to
£D.
Expressions (1.5a)-
(1.5d) give after rearrangement of the terms
AD=V2+
Y,
df+
J2 (xidj - Xjdi +
aij)2
+ J2 (XiV +
a0i)2
li3 lij3 1«3
= 2A -
m2
+ ^2
(ixidj
~
xjdi)2
+ 2{xidj - Xjdi)Gij +
cr2j)
+ 5Z (xi^Xi + x{D\xi, V] + XiVaoi + XiG^V + a^).
i
Using the fact that
[xi,V] = -707^ = -2cr0i,
(ay)2
= - - ,
(o-0i)2
= - ,
we obtain
AD = 2A -
m2
+ ] P ((xtfy -
Xjft)2
+ 2(x»9j - Xjdifaj - - )
+ ^ (x»(A - m2)xi + Xi[a0i,T} + - ) .
i
A direct calculation gives
[j0j,V] = imrf5 + ] T -(7j7z - lHj)di
1
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